Converting Decimal -3.11111111111111 to a Simplified Fraction

Step 1: Identify the Decimal

The decimal number you want to convert is -3.11111111111111. This number consists of a whole number part (-3) and a decimal part (.11111111111111).

Step 2: Separate the Whole Number from the Decimal

First, express the whole number part as a fraction. The whole number -3 can be written as -3/1.

Step 3: Convert the Decimal Part to a Fraction

The decimal .11111111111111 can be recognized as a repeating decimal, which is commonly written as 0.ar{1}. To convert this into a fraction, we can use the following method:

1. Let x = 0.ar{1}.
2. Multiply both sides of the equation by 10: 10x = 1.ar{1}.
3. Subtract the first equation from the second: 10x - x = 1.ar{1} - 0.ar{1}.
4. This gives you: 9x = 1.
5. Now, solve for x: x = 1/9.

Thus, 0.ar{1} = 1/9.

Step 4: Combine the Whole Number and the Fraction

Now, we can combine the whole number part and the fraction part:

-3.11111111111111 = -3 + 0.11111111111111 = -3 + 1/9 = -3 + 1/9 = -rac{27}{9} + rac{1}{9} = -rac{27 - 1}{9} = -rac{26}{9}.

Step 5: Final Result

So the decimal -3.11111111111111 as a simplified fraction is:

-rac{26}{9}.